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Uncertain Rule-Based Fuzzy Systems pp 529–616Cite as

Interval Type-2 Fuzzy Systems: Design Methods and Applications

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Abstract

This chapter is the IT2 version of Chap. 4. It focuses first on what exactly “design of an IT2 fuzzy system” means, and then provides a tabular way for making the choices that are needed in order to fully specify an IT2 fuzzy system. It introduces two approaches to design, the partially dependent approach and the totally independent approach, but this time for singleton, T1 non-singleton and IT2 non-singleton IT2 fuzzy systems. It then describes the extension of the six design methods that were covered for type-1 fuzzy systems in Chap. 4 to IT2 fuzzy systems, namely: IT2 WM, least squares, derivative-based, SVD-QR, derivative-free, and iterative. It continues the three Chap. 4 case studies (forecasting of time series, knowledge mining using surveys, and fuzzy logic control), and continues the Chap. 4 applications of forecasting of compressed video traffic using IT2 Mamdani and TSK fuzzy systems, and IT2 rule-based classification of video traffic. The application of equalization of time-varying nonlinear digital communication channels is also covered. Thirteen examples are used to illustrate the important concepts.

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Notes

  1. 1.

    See Sect. 6.9 for a definition of the quantities in (10.12).

  2. 2.

    Mendel (2004) is all about computing derivatives for IT2 Mamdani fuzzy system with COS type-reduction + defuzzification. Because it does not use the formulation of \( y({\mathbf{x}}) \) that is given in (9.172)–(9.176), but instead uses the formulation of \( y({\mathbf{x}}) \) that is in terms of the switch points \( L({\mathbf{x}}) \) and \( R({\mathbf{x}}) \), it is very complicated. This is due to having to reorder the IT2 FBFs, something that is not needed when (9.172)–(9.176) are used.

  3. 3.

    This is a somewhat speculative section because only the SVD-QR method that is described for the IT2 Mamdani fuzzy system with COS type-reduction + defuzzification has been successfully applied in Liang and Mendel (2000c), although Example 10.9’s Steps 4 and 5 are not explicitly mentioned in that paper, even though they were used.

  4. 4.

    If an IT2 fuzzy system cannot be designed by means of optimization, but can only be designed by trial and error, then it is possible that the performance of the singleton IT2 fuzzy system may be worse than that of a non-singleton type-1 fuzzy system, or the performance of the T1 non-singleton IT2 fuzzy system may be worse than that of a singleton IT2 fuzzy system, or the performance of the IT2 non-singleton IT2 fuzzy system may be worse than that of a T1 non-singleton IT2 fuzzy system. This occurs because it may not be possible to try all possible combinations of the design parameters, and is therefore the result of an incomplete design procedure. One must be very cautious about drawing conclusions from such trial and error designs.

  5. 5.

    In everything that has been described prior to this section the type-reduced set has only been a means to an end, the end being the defuzzified output. Although this author has also used the type-reduced set in this way, he always felt that it contained valuable information about dispersion about the output and that somehow this information ought to also be used in a design, since dispersion about the mean is used in probability based designs.

  6. 6.

    Cara et al. (2013) compare a singleton IT2 fuzzy system with a non-singleton T1 fuzzy system for nine function approximation problems to address a criticism of IT2 fuzzy systems that they have more parameters (design degrees of freedom) than a T1 fuzzy system. Both fuzzy systems are constrained to have the same number of parameters. The IT2 fuzzy system uses the FOU to handle input uncertainties, whereas the T1 fuzzy system uses non-singleton fuzzification to do this. The parameters of both kinds of fuzzy system are optimized. The paper demonstrates that the IT2 fuzzy system gives much better performance than the non-singleton T1 fuzzy system for high noise levels, and that it is not the use of extra parameters that lets a singleton IT2 fuzzy system outperform a non-singleton T1 fuzzy system, but rather their different ways of handing such uncertainty.

  7. 7.

    If the singular value threshold \( \gamma \) had been chosen to be smaller than 1, then the rule-reduced designs would have contained more rules, but would have achieved even smaller RMSEs than in the present designs.

  8. 8.

    Note that, for \( N\widetilde{V}L \), \( m_{a} = \sigma_{a} = 0 \), so the first two coordinates of both its UMF and LMF are (0, 0) and (0, 1), as they should be; and, for \( M\widetilde{A}A \), \( m_{b} = 10 \) and \( \sigma_{b} = 0 \), so the last two coordinates of both its UMF and LMF are (10, 1) and (10, 0), as they should be.

  9. 9.

    Note that \( \underline{C}_{avg}^{i} \) and \( \bar{C}_{avg}^{i} \) play the roles of \( c_{l} (\widetilde{G}^{i} ) \) and \( c_{r} (\widetilde{G}^{i} ) \), respectively.

  10. 10.

    See, also, Sect. 5.2.

  11. 11.

    Another approach for obtaining an IT2 FS word model from the same kind of data that are used by the IA, EIA and HMA is in Tahayori and Sadeghian (2012). Its Median Interval Approach (MIA) is based on calculating the median boundaries of the range of MFs associated with the words.

  12. 12.

    A T1 FS can be used either to model each subject’s intra-uncertainty, or to model the inter-uncertainty of a group of subjects (by using group statistics), but it cannot model both kinds of uncertainties simultaneously.

  13. 13.

    For those who are note interested in the specifics of this application, just view Fig. 4.17 as a time series, where “time” is the same as “frame index.”

  14. 14.

    Earlier it was argued that such a TSK fuzzy system is better referred to as a Mamdani fuzzy system; however, because there is no unnormalized Mamdani fuzzy system, but there is an unnormalized TSK fuzzy system, the designation “TSK” is used here.

  15. 15.

    The material in this section is taken from Karnik et al. (1999) and Liang and Mendel(2000b).

  16. 16.

    Definitions of the terms used in this section can be found in any standard textbook on communication theory, e.g. Proakis (1989).

  17. 17.

    Wikipedia/Adaptive equalizer: accessed on June 27, 2016.

  18. 18.

    Work that has been done in the area of adaptive equalization, but mainly for time-invariant channels, includes Proakis (1989) and the many references therein, and, Chen et al. (1993a, b, 1995), Cowan and Semnani (1998), Lee (1996), Moon and Jeon (1998), Patra and Mulgrew (1998), Sarwal and Srinath (1995), Savazzi et al. (1998), and Wang and Mendel (1993).

  19. 19.

    See footnote 14.

  20. 20.

    In the K-NNC, if the number of training prototypes is N, then \( K{ = }\sqrt N \) is the optimal choice for K. It is required that N be an odd integer; hence, the choice of \( N = 121 \).

  21. 21.

    Some of the material in this section is taken from Mendel et al. (2014, Chap. 1).

  22. 22.

    This section was prepared by Prof. Tufan Kumbasar.

  23. 23.

    KM algorithms were used to compute the COS type-reduced set; however, the same numerical results would have been obtained had the EKM or EAISC algorithms been used.

  24. 24.

    In (9.158), \( m = n = 1/2 \) and \( y^{i} \) equals the crisp consequents that are in Table 10.16.

  25. 25.

    Better performance may have been obtained for the three other IT2 FPID controllers if the parameters of their FOUs were tuned for each of them.

  26. 26.

    Some of the numbers in this table are very close to each other, and so it is advisable to use statistical tests [e.g., Garcia et al. (2010), Derrac et al. (2011)] to arrive at comparative conclusions. Additionally, in practice, results would be obtained for many more perturbed processes.

  27. 27.

    In these references the partitions are not referred to as IT2 first-, IT2 second- or IT2 novelty rule partitions. These names and explanations only occurred to this author during the writing of this book.

  28. 28.

    The statement of this exercise is taken or adapted for the most part from Mendel et al. (2014, Chap. 5, pp. 207–208).

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  1. University of Southern California, Los Angeles, CA, USA

    Jerry M. Mendel

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Mendel, J.M. (2017). Interval Type-2 Fuzzy Systems: Design Methods and Applications. In: Uncertain Rule-Based Fuzzy Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-51370-6_10

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